MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg2val Structured version   Unicode version

Theorem itg2val 19612
Description: Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypothesis
Ref Expression
itg2val.1  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2val  |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  ) )
Distinct variable group:    x, g, F
Allowed substitution hints:    L( x, g)

Proof of Theorem itg2val
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 xrltso 10726 . . 3  |-  <  Or  RR*
21supex 7460 . 2  |-  sup ( L ,  RR* ,  <  )  e.  _V
3 reex 9073 . 2  |-  RR  e.  _V
4 ovex 6098 . 2  |-  ( 0 [,]  +oo )  e.  _V
5 breq2 4208 . . . . . . 7  |-  ( f  =  F  ->  (
g  o R  <_ 
f  <->  g  o R  <_  F ) )
65anbi1d 686 . . . . . 6  |-  ( f  =  F  ->  (
( g  o R  <_  f  /\  x  =  ( S.1 `  g
) )  <->  ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
76rexbidv 2718 . . . . 5  |-  ( f  =  F  ->  ( E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) )  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
87abbidv 2549 . . . 4  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) } )
9 itg2val.1 . . . 4  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
108, 9syl6eqr 2485 . . 3  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  L )
1110supeq1d 7443 . 2  |-  ( f  =  F  ->  sup ( { x  |  E. g  e.  dom  S.1 (
g  o R  <_ 
f  /\  x  =  ( S.1 `  g ) ) } ,  RR* ,  <  )  =  sup ( L ,  RR* ,  <  ) )
12 df-itg2 19506 . 2  |-  S.2  =  ( f  e.  ( ( 0 [,]  +oo )  ^m  RR )  |->  sup ( { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) } ,  RR* ,  <  ) )
132, 3, 4, 11, 12fvmptmap 7042 1  |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   {cab 2421   E.wrex 2698   class class class wbr 4204   dom cdm 4870   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Rcofr 6296   supcsup 7437   RRcr 8981   0cc0 8982    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113   [,]cicc 10911   S.1citg1 19499   S.2citg2 19500
This theorem is referenced by:  itg2cl  19616  itg2ub  19617  itg2leub  19618  itg2addnclem  26246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-itg2 19506
  Copyright terms: Public domain W3C validator